Question

Suppose that 9 green balls and 13 purple balls are placed in an urn. Two balls are then drawn in succession. What is the probability that both balls drawn have the same color if the first ball is replaced before the second is drawn?

Answer 1

The probability of drawing a green ball first is 9/22 and the probability of drawing another green ball next is 9/22. Therefore the probability of drawing two green balls in succession is
\[P(2\ green\ successively)=\frac{9}{22}\times\frac{9}{22}\]
By the same reasoning the probability of drawing two purple balls in succession is
\[P(2\ purple\ successively)=\frac{13}{22}\times\frac{13}{22}\]
The events 'draw two green successively' and 'draw two purple successively' are muatully exclusive, therefore the probability that both balls draw successively have the same color is
\[(\frac{9}{22}\times\frac{9}{22})+(\frac{13}{22}\times\frac{13}{22})=you\ can\ calculate\]

Answer 2

So I wouldn't need to draw a tree diagram for this? lol

Answer 3

mutually exclusive*

Answer 4

It was just the last part of the question that threw me off. If it was replaced. That one

Answer 5

So if the two balls are still drawn successively, but the second ball drawn is a purple one. Do I do the same thing but instead I multiply another purple?

Answer 6

If the ball drawn first was green and it was not replaced, the probability of drawing two green balls in succession is
\[P(2\ green\ without\ replacement)=\frac{9}{22}\times\frac{8}{22}\]
Do you follow?

Answer 7

yes

Answer 8

nvm, it's not because the first ball is replaced correct?

Answer 9

^regarding my second question

Answer 10

It is not necessary to find the probability of drawing two different colored balls in succession. Introducing this probability would unnecessarily complicate finding the solution.

Answer 11

nvm i'm confusing myself. sorry lol

Answer 12

No problem :)

Answer 13

so if the second ball i drew was purple, and the first ball is replaced before the second (like the first situation), you're saying that there is not need to do anything different correct?

Answer 14

i just need to find the probability of getting that purple ball

Answer 15

?

Answer 16

maybe writing out in a formula is better for me. I just noticed that... lol. The P(A/B) equation

Answer 17

You just need to find the probabilities of the two events 'A: Draw two green balls successively, with replacement' and 'B: Draw two purple balls successively, with replacement'. Then the probability of the event 'C: Draw two balls of the same color successively, with replacement' is the sum of the probabilities of events A and B.
\[P(C)=P(A \cup B)=P(A)+P(B)\]